Element structure of general affine group of degree two over a finite field

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This article gives specific information, namely, element structure, about a family of groups, namely: general affine group of degree two.
View element structure of group families | View other specific information about general affine group of degree two

This article gives the element structure of the general affine group of degree two over a finite field. Similar structure works over an infinite field or a field of infinite characteristic, with suitable modification. For more on that, see element structure of general affine group of degree two over a field.

The discussion here builds upon the discussion of element structure of general linear group of degree two over a finite field.

Summary

Item Value
order q^2(q^2 - 1)(q^2 - q) = q^3(q + 1)(q - 1)^2
exponent  ?
number of conjugacy classes q^2 + q - 1

Particular cases

q
(field size)
p
(underlying prime, field characteristic)
r = \log_pq general affine group GL(2,q) order of the group (= q^2(q^2 - 1)(q^2 - q)) number of conjugacy classes (= q^2 + q - 1) element structure page
2 2 1 symmetric group:S4 24 5 element structure of symmetric group:S4
3 3 1 general affine group:GA(2,3) 432 11 element structure of general affine group:GA(2,3)
4 2 2 general affine group:GA(2,4) 2880 19
5 5 1 general affine group:GA(2,5) 12000 29

Conjugacy class structure


There is a total of q^2(q^2 - 1)(q^2 - q) = q^3(q - 1)^2(q + 1) elements, and there are q^2 + q - 1 conjugacy classes of elements.

The conjugacy class structure is closely related to that of GL(2,q) -- see Element structure of general linear group of degree two over a finite field#Conjugacy class structure.

We describe a generic element of GA(2,q) in the form:

x \mapsto Ax + v

where A \in GL(2,q) is the dilation component and v \in (\mathbb{F}_q)^2 is the translation component.

Consider the quotient mapping GA(2,q) \to GL(2,q), which sends the generic element to A. Under this mapping, the following is true:

  • For those conjugacy classes of GL(2,q) comprising elements that do not have 1 as an eigenvalue, the full inverse image of the conjugacy class is a single conjugacy class in GA(2,q). In other words, the translation component does not matter.
  • For those conjugacy classes of GL(2,q) comprising elements that do have 1 as an eigenvalue, the conjugacy class splits into two depending on whether v is in the image of (A - 1).
Nature of conjugacy class Eigenvalues Characteristic polynomial of A Minimal polynomial of A Size of conjugacy class Number of such conjugacy classes Total number of elements Is A semisimple? Is A diagonalizable over \mathbb{F}_q?
A is the identity, v = 0 \{ 1,1 \} (x - 1)^2 x - 1 1 1 1 Yes Yes
A is the identity, v \ne 0 \{ 1,1 \} (x - 1)^2 x - 1 q^2 - 1 1 q^2 - 1 Yes Yes
A is diagonalizable over \mathbb{F}_q with equal diagonal entries not equal to 1, hence a scalar. The value of v does not affect the conjugacy class. \{a,a \} where a \in \mathbb{F}_q^\ast \setminus \{ 1 \} (x - a)^2 where a \in \mathbb{F}_q^\ast \setminus \{ 1 \} x - a where a \in \mathbb{F}_q^\ast \setminus \{ 1 \} q^2 q - 2 q^2(q - 2) Yes Yes
A is diagonalizable over \mathbb{F}_{q^2}, not over \mathbb{F}_q. Must necessarily have no repeated eigenvalues. The value of v does not affect the conjugacy class. Pair of conjugate elements of \mathbb{F}_{q^2} x^2 - ax + b, irreducible Same as characteristic polynomial q^3(q - 1) q(q - 1)/2 = (q^2 - q)/2 q^4(q-1)^2/2 Yes No
A has Jordan block of size two, with repeated eigenvalue equal to 1, v is in the image of A - 1 \{ 1, 1 \} (x - 1)^2 Same as characteristic polynomial q(q^2 - 1) 1 q(q^2 - 1) No No
A has Jordan block of size two, with repeated eigenvalue equal to 1, v is not in the image of A - 1 \{ 1, 1 \} (x - 1)^2 Same as characteristic polynomial (q^2 - 1)(q^2 - q) 1 (q^2 - 1)(q^2 - q) No No
A has Jordan block of size two, with repeated eigenvalue not equal to 1 a (multiplicity two) where a \in \mathbb{F}_q^\ast \setminus \{ 1 \} (x - a)^2 where a \in \mathbb{F}_q^\ast \setminus \{ 1 \} Same as characteristic polynomial q^2(q^2 - 1) q - 2 q^2(q^2 - 1)(q - 2) No No
A diagonalizable over \mathbb{F}_q with distinct diagonal entries, one of which is 1, v is in the image of A - 1 1,\mu, \mu \in \mathbb{F}_q^\ast \setminus \{ 1 \} x^2 - (\mu + 1)x + \mu Same as characteristic polynomial q^2(q + 1) q - 2 q^2(q+1)(q-2) Yes Yes
A diagonalizable over \mathbb{F}_q with distinct diagonal entries, one of which is 1, v is not in the image of A - 1 1,\mu, \mu \in \mathbb{F}_q^\ast \setminus \{ 1 \} x^2 - (\mu + 1)x + \mu Same as characteristic polynomial q(q + 1)(q^2 - q) q - 2 q^2(q+1)(q - 1)(q-2) Yes Yes
A diagonalizable over \mathbb{F}_q with distinct diagonal entries, neither of which is 1 \lambda, \mu (interchangeable) distinct elements of \mathbb{F}_q^\ast, neither equal to 1 x^2 - (\lambda + \mu)x + \lambda \mu Same as characteristic polynomial q^3(q+1) (q - 2)(q - 3)/2 q^3(q+1)(q-2)(q-3)/2 Yes Yes
Total NA NA NA NA q^2 + q - 1 q^2(q^2 - 1)(q^2 - q)